Analytical scanning evanescent microwave microscope and control stage

ABSTRACT

A scanning evanescent microwave microscope (SEMM) that uses near-field evanescent electromagnetic waves to probe sample properties is disclosed. The SEMM is capable of high resolution imaging and quantitative measurements of the electrical properties of the sample. The SEMM has the ability to map dielectric constant, loss tangent, conductivity, electrical impedance, and other electrical parameters of materials. Such properties are then used to provide distance control over a wide range, from to microns to nanometers, over dielectric and conductive samples for a scanned evanescent microwave probe, which enable quantitative non-contact and submicron spatial resolution topographic and electrical impedance profiling of dielectric, nonlinear dielectric and conductive materials. The invention also allows quantitative estimation of microwave impedance using signals obtained by the scanned evanescent microwave probe and quasistatic approximation modeling. The SEMM can be used to measure electrical properties of both dielectric and electrically conducting materials.

CLAIM OF PRIORITY

This application is a division of U.S. patent application Ser. No.09/608,311 filed Jun. 30, 2000, now U.S. Pat. No. 7,550,963 issued Jun.23, 2009, which claims the benefit of U.S. Provisional Application No.60/141,698 filed Jun. 30, 1999, and is a continuation-in-part of U.S.patent application Ser. No. 09/158,037 filed Sep. 22, 1998, now U.S.Pat. No. 6,173,604 issued Jan. 16, 2001, which claims the benefit ofU.S. Provisional Application No. 60/059,471, filed Sep. 22, 1997 andwhich is a continuation-in-part of U.S. patent application Ser. No.08/717,321 filed Sep. 20, 1996, now U.S. Pat. No. 5,821,410 issued Oct.13, 1998.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with U.S. Government support under Contract No.DE-AC03-76SF00098 between the U.S. Department of Energy and theUniversity of California for the operation of Lawrence BerkeleyLaboratory. The U.S. Government may have certain rights in thisinvention.

INCORPORATION-BY-REFERENCE OF MATERIAL SUBMITTED ON A COMPACT DISC

Not Applicable

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to scanning probe microscopy and morespecifically to scanning evanescent electromagnetic wave microscopyand/or spectroscopy.

2. Description of Related Art

Quantitative dielectric measurements are currently performed by usingdeposited electrodes on large length scales (mm) or with a resonantcavity to measure the average dielectric constant of the specimen beingtested. Quantitative conductivity measurements of the test specimen canonly be accurately performed with a four-point probe. A drawbackassociated with performing the aforementioned measurements is that theprobe tip used to measure the dielectric and conductivity properties ofthe test specimen often comes into contact with the test specimen.Repeated contact between the probe tip and the test specimen causesdamage to both the probe tip and the specimen, thereby making theresulting test measurements unreliable.

Another drawback associated with the aforementioned measurements is thatgap distance between the probe tip and the test specimen can only beaccurately controlled over a milli-meter (mm) distance range.

BRIEF SUMMARY OF THE INVENTION

The present invention allows quantitative non-contact andhigh-resolution measurements of the complex dielectric constant andconductivity at RF or microwave frequencies. The present inventioncomprises methods of tip-sample distance control over dielectric andconductive samples for the scanned evanescent microwave probe, whichenable quantitative non-contact and high-resolution topographic andelectrical impedance profiling of dielectric, nonlinear dielectric andconductive materials. Procedures for the regulation of the tip-sampleseparation in the scanned evanescent microwave probe for dielectric andconducting materials are also provided.

The present invention also provides methods for quantitative estimationof microwave impedance using signals obtained by scanned evanescentmicrowave probe and quasistatic approximation modeling. The applicationof various quasistatic calculations to the quantitative measurement ofthe dielectric constant, nonlinear dielectric constant, and conductivityusing the signal from a scanned evanescent microwave probe are provided.Calibration of the electronic system to allow quantitative measurements,and the determination of physical parameters from the microwave signalis also provided.

The present invention also provides methods of fast data acquisition ofresonant frequency and quality factor of a resonator; more specifically,the microwave resonator in a scanned evanescent microwave probe.

A piezoelectric stepper for providing coarse control of the tip-sampleseparation in a scanned evanescent microwave probe with nanometer stepsize and centimeter travel distances is disclosed.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)

FIGS. 1(A)-1(B) are schematic views of equivalent circuits used formodeling the tip-sample interaction;

FIG. 2 is a schematic view of the tip-sample geometry;

FIG. 3 illustrates the infinite series of image charges used todetermine the tip-sample impedance;

FIG. 4 is a graph showing the agreement between the calculated andmeasure frequency shifts for variation of the tip-sample separation.

FIG. 5 illustrates the calculation of C_(tip-sample);

FIG. 6 is a schematic view of the setup used to measure the nonlineardielectric constant;

FIG. 7 shows images of topography and ∈₃₃₃ for a periodically poledsingle-crystal LiNbO₃ wafer;

FIG. 8 is a graph showing f_(r) and Δ(1/Q) as a function ofconductivity;

FIG. 9 illustrates the tip-sample geometry modeled;

FIG. 10 is a graph showing C_(r) using the model approximation;

FIG. 11 is a schematic view of the operation of the microscope forsimultaneous measurement of the topography and nonlinear dielectricconstant;

FIG. 12 shows images of topography and ∈₃₃₃ for a periodically poledsingle crystal LiNb0₃ wafer;

FIG. 13 is a graph for the C_(tip-sample) calculation;

FIG. 14( a)-14(b) are two graphs showing the variation of the derivativesignal versus tip-sample variation;

FIG. 15 is a graph showing calibration for regulation of tip-sampleseparation;

FIG. 16 illustrates the measurement of topographic and resistivityvariations;

FIG. 17 is a schematic view of the architecture of the data acquisitionand control electronics;

FIG. 18 is a flow chart for of the architecture of the inventive dataacquisition and control electronics;

FIG. 19 illustrates the design and operation of the piezoelectricstepper;

FIG. 20 illustrates the sequence of motion of the piezoelectric stepper;

FIG. 21 illustrates the integration of an AFM tip with SEMM.

DETAILED DESCRIPTION OF THE INVENTION

Embodiment A

To determine quantitatively the physical properties, such as the complexdielectric constant, nonlinear dielectric constant and conductivity,through measurements of changes in resonant frequency (f_(r)) andquality factor (Q) as function of different materials, bias electric andmagnetic fields, tip-sample distance and temperature, etc. by a scannedevanescent microwave probe (SEMP), a quantitative model of the electricand magnetic fields in the tip-sample interaction region is necessary. Anumber of quasistatic models can be applied to the calculation of theprobe response to dielectric, nonlinear dielectric and conductivematerials. For the present invention, these models are applied to thecalculation of the complex dielectric constant, nonlinear dielectricconstant, and conductivity.

To determine the electrical properties of a sample, the variation inresonant frequency (f_(r)) and quality factor (Q) of a resonant cavityis measured. (FIG. 1(A)). The tip-sample interaction is modeled usingthe equivalent RLC circuit shown in FIG. 1(B). FIG. 1(A) shows the novelevanescent probe structure comprising a microwave resonator such asillustrated microwave cavity 10 with coupling loops for signal input andoutput. The sharpened metal tip 20, which, in accordance with theinvention acts as a point-like evanescent field emitter as well as adetector, extends through a cylindrical opening or aperture 22 inendwall 16 of cavity 10. Mounted immediately adjacent sharpened tip 20is a sample 80. Cavity 10 comprises a standard quarter or half wavecylindrical microwave cavity resonator having a central metal conductor18 with a tapered end 19 to which is attached sharpened metal tip orprobe 20. The tip-sample interaction appears as an equivalent complextip-sample capacitance (C_(tip-sample)). Given f_(r) and Q, the complextip-sample capacitance can be extracted.

$\begin{matrix}{\frac{\Delta\; f}{f_{0}} = {- \frac{C_{r}}{2C}}} & (1) \\{{\Delta\left( \frac{1}{Q\;} \right)} = {{- \left( {\frac{1}{Q_{0}} + \frac{2C_{i}}{C_{R}\;}} \right)}\frac{\Delta\; f}{f_{0\;}}}} & (2)\end{matrix}$where

${C_{{tip}\text{-}{sample}} = {C_{r} + {{\mathbb{i}}\; C_{I}}}},{{\Delta\; f} = {f_{r} - f_{0}}},{{\Delta\left( \frac{1}{Q} \right)} = {\frac{1}{Q} - \frac{1}{Q_{0}}}},$and f₀ and Q₀ are the unloaded resonant frequency and quality factor.The calculation of C_(tip-sample) is described in greater detail below.

a) Modeling of the Cavity Response for Dielectric Materials

To allow the quantitative calculation of the cavity response to a samplewith a certain dielectric constant, a detailed knowledge of the electricand magnetic fields in the probe region is necessary. FIG. 2 describesthe tip-sample geometry. The most general approach is to apply an exactfinite element calculation of the electric and magnetic fields for atime-varying three dimensional region. This is difficult and timeconsuming, particularly for the tip-sample geometry described in FIG. 2.Since the tip is sharply curved, a sharply varying mesh size should beimplemented. Since the spatial extent of the region of the sample-tipinteraction is much less than the wavelength of the microwave radiationused to probe the sample (λ˜28 cm at 1 GHz, ˜14 cm at 2 GHz), thequasistatic approximation can be used, i.e. the wave nature of theelectric and magnetic fields can be ignored. This allows the relativelyeasy solution of the electric fields inside the dielectric sample. Afinite element calculation of the electric and magnetic fields underquasistatic approximation for the given tip-sample geometry can beapplied. A number of other approaches can be employed for thedetermination of the cavity response with an analytic solution, whichare much more convenient to use. The calculation of the relation betweencomplex dielectric constant and SEMP signals for bulk and thin filmdielectric materials by means of an image charge approach is nowoutlined.

Spherical Tip

i. Calculation of the Complex Dielectric Constant for Bulk Materials.

The complex dielectric constant measured can be determined by an imagecharge approach if signals from the SEMP are obtained. By modeling theredistribution of charge when the sample is brought into the proximityof the sample, the complex impedance of the sample for a giventip-sample geometry can be determined with measured cavity response(described below). A preferred model is one that can have an analyticexpression for the solution and is easily calibrated and yieldsquantitatively accurate results. Since the tip geometry will varyappreciably between different tips, a model with an adjustable parameterdescribing the tip is required. Since the region close to the tippredominately determines the sample response, the tip as a metal sphereof radius R₀ can be modeled.

FIG. 3 illustrates the infinite series of image charges used todetermine the tip sample impedance. For dielectric samples, thedielectric constant is largely real

$\left( {\frac{ɛ_{i}}{ɛ_{r\;}} < 0.1} \right),$where ∈_(r) and ∈_(i) are the real and imaginary part of the dielectricconstant of the sample, respectively. Therefore, the real portion of thetip-sample capacitance, C_(r), can be calculated directly and theimaginary portion of the tip-sample capacitance, C_(i), can becalculated by simple perturbation theory.

Using the method of images, the tip-sample capacitance is calculated bythe following equation:

$\begin{matrix}{C_{r} = {4\pi\; ɛ_{0}R_{0}{\sum\limits_{n = 1}^{\infty}\frac{{bt}_{n}}{a_{1} + a_{n}}}}} & (3)\end{matrix}$where t_(n) and a_(n) have the following iterative relationships:

$\begin{matrix}{a_{n} = {1 + a^{\prime} - \frac{1}{1 + a^{\prime} + a_{n - 1}}}} & (4) \\{t_{n} = \frac{{bt}_{n - 1}}{1 + a^{\prime} + a_{n - 1}}} & (5)\end{matrix}$with

${a_{1} = {1 + a^{\prime}}},{t_{1} = 1},{b = \frac{ɛ - ɛ_{0}}{ɛ + ɛ_{0}}},{{{and}\mspace{14mu} a^{\prime}} = \frac{d}{R}},$where ∈ is the dielectric constant of the sample, ∈₀ is the permittivityof free space, d is the tip-sample separation, and R is the tip radius.

This simplifies to:

$\begin{matrix}{C_{r} = {4\pi\; ɛ_{0}{R_{0}\left\lbrack {\frac{\ln\left( {1 - b} \right)}{b} + 1} \right\rbrack}}} & (6)\end{matrix}$as the tip-sample gap approaches zero.

Since the dielectric constant of dielectric materials is primarily real,the loss tangent (tan δ) of dielectric materials can be determined byperturbation theory. The imaginary portion of the tip-sample capacitancewill be given by:C_(i)=C_(r) tan δ.  (7)

Given the instrument response, C_(tip-sample), the complex tip-samplecapacitance, and therefore the complex dielectric constant of the samplecan be estimated.

FIG. 4 illustrates agreement between the calculated and measurefrequency shifts for variation of the tip-sample separation. Thedielectric constant and loss tangent can be determined fromC_(tip-sample) by a number of methods. One simple approach is toconstruct a look-up table which yields the dielectric constantcorresponding to a given C_(tip-sample). Alternatively, it can bedirectly calculated from the signals using the formula. Table 1 shows acomparison between measured and reported values for dielectric constantand loss tangent.

The perturbed electric field inside the sample is:

$\begin{matrix}{{{\overset{\rightarrow}{E}}_{1}\left( {ɛ,d} \right)} = {\frac{q}{2{\pi\left( {ɛ + ɛ_{0}} \right)}}{\sum\limits_{n = 1}^{\infty}{t_{n}\frac{{r{\overset{\rightarrow}{e}}_{r}} + {\left( {z + a_{n}} \right){\overset{\rightarrow}{e}}_{z}}}{\left\lbrack {r^{2} + \left( {z + a_{n}} \right)^{2}} \right\rbrack^{3/2}}}}}} & (8)\end{matrix}$where q=4π∈₀RV₀, V₀ is the voltage, and {right arrow over (e)}_(r) and{right arrow over (e)}_(z). are the unit vectors along the directions ofthe cylindrical coordinates r and z, respectively.

ii. Calculation of the Complex Dielectric Constant for Thin Films.

The image charge approach can be adapted to allow the quantitativemeasurement of the dielectric constant and loss tangent of thin films.FIG. 5 illustrates the calculation of C_(tip-sample). In the strictsense, the image charge approach will not be applicable to thin filmsdue to the divergence of the image charges shown in FIG. 5. However, ifthe contribution of the substrate to the reaction on the tip can bemodeled properly, the image charge approach is still a goodapproximation. According to the present invention, it is expected thatall films can be considered as bulk samples if the tip is sharp enoughsince the penetration depth of the field is only about R. Thecontribution from the substrate will decrease with increases in filmthickness and dielectric constant. This contribution can be modeled byreplacing the effect of the reaction from the complicated image chargeswith an effective charge with the following format:

$\begin{matrix}{{b_{eff} = {b_{20} + {\left( {b_{10} - b_{20}} \right){\exp\left( {{- 0.18}\;\frac{a}{1 - b_{20}}} \right)}}}},} & (9)\end{matrix}$where

${b_{20} = \frac{ɛ_{2} - ɛ_{1}}{ɛ_{2} + ɛ_{1}}},{b_{10} = \frac{ɛ_{1} - ɛ_{0}}{ɛ_{1} + ɛ_{0}}},$∈₂ and ∈₁ are the dielectric constants of the film and substrate,respectively,

${a = \frac{d}{R}},$and d is the thickness of the film. This format reproduces the thin andthick film limits for the signal. The constant 0.18 was obtained bycalibrating against interdigital electride measurements at the samefrequency on SrTiO₃ thin film. Following a similar process to theprevious derivation, yields:

$\begin{matrix}{{C_{r} = {4\pi\; ɛ_{0}R{\sum\limits_{n = 1}^{\infty}{\sum\limits_{m = 0}^{\infty}{b_{eff}^{n - 1}b_{21}^{m}{b_{10}^{m}\left( {\frac{b_{20}}{n + 1 + {2{mna}}} - \frac{b_{21}}{n + 1 + {2\left( {m + 1} \right){na}}}} \right)}}}}}},} & (10) \\{{C_{I} = {4\pi\; ɛ_{0}R{\sum\limits_{n = 1}^{\infty}{\sum\limits_{m = 0}^{\infty}{b_{eff}^{n - 1} b_{21}^{m}{b_{10}^{m}\left( \begin{matrix}{{\tan\;\delta_{2}\frac{b_{20}}{n + 1 + {2{mna}}}} - \frac{b_{21}}{\left( {n + 1 + {2\left( {m + 1} \right){na}}} \right)} +} \\{\frac{2ɛ_{1}ɛ_{2}\tan\;\delta_{1}}{\left( {ɛ_{2} + ɛ_{1}} \right)\left( {ɛ_{2} + ɛ_{0}} \right)}\frac{b_{21}}{\left( {n + 1 + {2\left( {m + 1} \right){na}}} \right)}}\end{matrix} \right)}}}}}},} & (11)\end{matrix}$where

${b_{21} = \frac{ɛ_{2} - ɛ_{1}}{ɛ_{2} + ɛ_{1}}},$tan ∈₂ and ∈₁ are the tangent losses of the film and substrate. Table 2lists the results of thin film measurements using the SEMP andinterdigital electrodes at the same frequency (1 GHz).

b) Calculation of the Nonlinear Dielectric Constant

The detailed knowledge of the field distribution in Eqn. 8 allowsquantitative calculation of the nonlinear dielectric constant. Thecomponent of the electric displacement D perpendicular to the samplesurface is given by:D ₃ =P ₃+∈₃₃(E _(l) +E _(m))+½∈₃₃₃(E _(l) +E _(m))²+⅙∈₃₃₃₃(E _(l) +E_(m))³+ . . .  (12)where D₃ is the electric displacement perpendicular to the samplesurface, P₃ is the spontaneous polarization, ∈_(ij), ∈_(ijk), ∈_(ijkl),. . . are the second-order (linear) and higher order (nonlinear)dielectric constants, respectively.

Since the field distribution is known for a fixed tip-sample separation,an estimate of the nonlinear dielectric constant from the change inresonance frequency with applied voltage can be made. For tip-sampleseparations much less than R, the signal mainly comes from a smallregion under the tip where the electric fields (both microwave electricfield E_(m) and low frequency bias electric field E_(l)) are largelyperpendicular to the sample surface. Therefore, only the electric fieldperpendicular to the surface needs to be considered.

From Eqn. 12, the effective dielectric constant with respect to E_(m)can be expressed as a fraction of E₁:

$\begin{matrix}{{{ɛ_{33}\left( E_{l} \right)} = {\frac{\partial D_{3}}{\partial E_{m}} = {ɛ_{33} + {ɛ_{333}\left( {E_{l} + E_{m}} \right)} + {\frac{1}{2}{ɛ_{3333}\left( {E_{l} + E_{m}} \right)}^{2}} + \ldots}}}\mspace{20mu},} & (13)\end{matrix}$and the corresponding dielectric constant change caused by E_(l) is:

$\begin{matrix}{{\Delta\; ɛ} = {{ɛ_{333}E_{l}} + {\frac{1}{2}ɛ_{3333}E_{l}^{2}} + \ldots}} & (14)\end{matrix}$

The change in f_(r) for a given applied electric field E_(l) is relatedto the change in the energy stored in the cavity. Since the electricfield for a given dielectric constant is known and the change in thedielectric constant is small, this can be calculated by integrating overthe sample:

$\begin{matrix}{\frac{{df}_{r}}{f_{r}} = {{- \frac{\int_{V_{s}}{\Delta\; ɛ\; E_{m}^{2}{\mathbb{d}v}}}{\int_{V_{1}}{\left( {{ɛ\; E_{0}^{2}} + {\mu\; H_{0}^{2}}} \right){\mathbb{d}v}}}} = {- \frac{\int_{V_{s}}{\left( {{ɛ_{333}E_{l}} + {\frac{1}{2}ɛ_{3333}E_{l}^{2}} + \ldots} \right)E_{m}^{2}{\mathbb{d}V}}}{\int_{V_{i}}{\left( {{ɛ\; E_{0}^{2}} + {\mu\; H_{0}^{2}}} \right){\mathbb{d}V}}}}}} & (15)\end{matrix}$where V_(s) is the volume of the sample containing electric field, H₀ isthe microwave magnetic field, and V_(t) is the total volume containingelectric and magnetic fields, possibly with dielectric filling ofdielectric constant ∈. E_(m) is given by Eqn. 8. The application of abias field requires a second electrode located at the bottom of thesubstrate. If the bottom electrode to tip distance is much larger thantip-sample distance and tip radius, Eqn. 8 should also hold for E_(l).The upper portion can be calculated by integrating the resultingexpression. The lower portion of the integral can be calibrated bymeasuring the dependence of f_(r) versus the tip-sample separation for abulk sample of known dielectric constant. If the tip-sample separationis zero, the formula can be approximated as:

$\begin{matrix}{{C_{r}(V)} = {{C_{r}\left( {V = 0} \right)} + {4\pi\; ɛ_{0}R\;\frac{1}{32}\frac{A}{ɛ_{33}}\frac{V}{R}\frac{ɛ_{33} + ɛ_{0}}{2ɛ_{0}}ɛ_{333}}}} & (16)\end{matrix}$where V is the low frequency voltage applied to the tip. Thiscalculation can be generalized in a straightforward fashion to considerthe effects of other nonlinear coefficients and thin films.

FIG. 6 illustrates the setup used to measure the nonlinear dielectricconstant. To measure ∈₃₃₃, an oscillating voltage V_(Ω), of frequencyf_(Ω), is applied to the silver backing of the sample and the output ofthe mixer is monitored with a lock-in amplifier (SR 830). This biasvoltage will modulate the dielectric constant of a nonlinear dielectricmaterial at f_(Ω). By measuring f_(r) and the first harmonic variationin the phase output simultaneously, sample topography and ∈₃₃₃ can bemeasured simultaneously.

FIG. 7 shows images of topography and ∈₃₃₃ for a periodically poledsingle-crystal LiNbO₃ wafer. The crystal is a 1 cm×1 cm single crystalsubstrate, poled by periodic variation of dopant concentration. Thepoling direction is perpendicular to the plane of the substrate. Thedielectric constant image is essentially featureless, with the exceptionof small variations in dielectric constant correlated with the variationin dopant concentration. The nonlinear image is constructed by measuringthe first harmonic of the variation in output of the phase detectorusing a lock-in amplifier. Since ∈_(ijk) reverses when the polarizationswitches, the output of the lock-in switches sign when the domaindirection switches. The value (−2.4×10⁻¹⁹ F/V) is within 20% of bulkmeasurements. The nonlinear image clearly shows the alternating domains.

c) Calculation of the Conductivity

i. Low Conductivity

The dielectric constant of a conductive material at a given frequency fmay be written as:

${ɛ = {ɛ_{r} + \frac{2{\mathbb{i}\sigma}}{f}}},$where ∈_(r), is the real part of the permittivity and σ is theconductivity. The quasistatic approximation should be applicable whenthe wavelength inside the material is much greater (>>) than thetip-sample geometry. For R₀˜1 um and λ=14 cm,

$\begin{matrix}{{ɛ_{\max} \approx \left( \frac{\lambda}{R_{0}} \right)^{2} \approx {2 \times 10^{10}}}{or}{{\sigma_{\max} \approx \frac{f\; ɛ_{\max}}{2} \approx {2 \times 10^{7}\frac{1}{\Omega - {cm}}}},}} & (17)\end{matrix}$

For σ<<σ_(max), the quasistatic approximation remains valid.C_(tip-sample) can be calculated by the method of images. Each imagecharge will be out of phase with the driving voltage. By calculating thecharge (and phase shift) accumulated on the tip when it is driven by avoltage V, frequency f, one can calculate a complex capacitance. Formoderate tip-sample separations, it is primarily a capacitance with asmaller real component.

Using the method of images, we find that the tip-sample capacitance isgiven by:

$\begin{matrix}{{C_{{tip} - {sample}} = {4\;\pi\; ɛ_{0}R_{0}{\sum\limits_{n = 1}^{\infty}\;\frac{{bt}_{n}}{a_{1} + a_{n}}}}},} & (18)\end{matrix}$where t_(n) and a_(n) have the following iterative relationships:

$\begin{matrix}{a_{n} = {1 + a^{\prime} - \frac{1}{1 + a^{\prime} + a_{n - 1}}}} & (19) \\{t_{n} = \frac{{bt}_{n - 1}}{1 + a^{\prime} + a_{n - 1}}} & (20)\end{matrix}$with

${a_{1} = {1 + a^{\prime}}},{t_{1} = 1},{b = \frac{ɛ - ɛ_{0}}{ɛ + ɛ_{0}}},{{{and}\mspace{14mu} a^{\prime}} = \frac{d}{R}},$where ∈=∈_(r)−i∈_(i) is the complex dielectric constant of the sample,∈_(o) is the ₀ permittivity of free space, d is the tip-sampleseparation, and R is the tip radius.

By expressing b=b_(r)+ib_(i)=|b|e^(iφ), the real and complexcapacitances can be separated.

$\begin{matrix}{{C_{r} = {4\;\pi\; ɛ_{0}R_{0}{\sum\limits_{n = 1}^{\infty}\;\frac{{b}^{n}{\cos\left( {n\;\varphi} \right)}g_{n}}{a_{1} + a_{n}}}}}{and}} & (21) \\{{C_{i} = {4\;\pi\; ɛ_{0}R_{0}{\sum\limits_{n = 1}^{\infty}\;\frac{{b}^{n}{\sin\left( {n\;\varphi} \right)}g_{n}}{a_{1} + a_{n}}}}},} & (22)\end{matrix}$where g₁=1 and g_(n) is given by:

$\begin{matrix}{g_{n} = \frac{g_{n - 1}}{1 + a^{\prime} + a_{n - 1}}} & (23)\end{matrix}$This calculation can be generalized in a straightforward fashion to thinfilms.

FIG. 8 illustrates f_(r) and Δ (1/Q) as a function of conductivity. Thecurve peaks approximately where the imaginary and real components of sbecome equal. For those plots, it is assumed that the complex dielectricconstant was given by

$ɛ = {10 + {{\mathbb{i}}{\frac{2\;\sigma}{f}.}}}$

ii. High Conductivity

For σ>σ_(max), the magnetic field should also be considered. The realportion of C_(tip-sample) can be derived using an image charge approach.This is identical to letting b=1.

$\begin{matrix}{{C_{{tip} - {sample}} = {4\;\pi\; ɛ_{0}R_{0}{\sum\limits_{n = 1}^{\infty}\;\frac{t_{n}}{a_{1} + a_{n}}}}},} & (24)\end{matrix}$where t_(n) and a_(n) have the following iterative relationships:

$\begin{matrix}{{a_{n} = {1 + a^{\prime} - \frac{1}{1 + a^{\prime} + a_{n - 1}}}}{and}} & (25) \\{t_{n} = \frac{t_{n - 1}}{1 + a^{\prime} + a_{n - 1}}} & (26)\end{matrix}$with

${a_{1} = {1 + a^{\prime}}},{t_{1} = 1},{b = \frac{ɛ - ɛ_{0}}{ɛ + ɛ_{0}}},{{{and}\mspace{14mu} a^{\prime}} = \frac{d}{R}},$where ∈ is the dielectric constant of the sample, ∈₀ is the permittivityof free space, d is the tip-sample separation, and R is the tip radius.

In this limit, the formula can also be reduced to a sum of hyperbolicsines; see E. Durand, Electrostatique, 3 vol. (1964-66).

$\begin{matrix}{C_{r} = {4\;\pi\; ɛ_{0}R_{0}\sinh\; a{\sum\limits_{n = 2}^{\infty}\;\frac{1}{\sinh\;{na}}}}} & (27)\end{matrix}$where a=cosh⁻¹(1+a′)

The magnetic and electric fields at the surface of the conductor aregiven by:

$\begin{matrix}{{{\overset{\rightarrow}{E}}_{s}(r)} = {\frac{R_{0}}{2\;\pi\; ɛ_{0}}{\sum\limits_{n = 1}^{\infty}{\frac{a_{n}^{\prime}q_{n}}{\left\lbrack {r^{2} + \left( {a_{n}^{\prime}R_{0}} \right)^{2}} \right\rbrack^{3/2}}{\overset{\rightarrow}{e}}_{z}}}}} & (28) \\{{{\overset{\rightarrow}{H}}_{s}(r)} = {{- {\mathbb{i}}}\frac{\omega}{2\;\pi\; r}{\sum\limits_{n = 1}^{\infty}{q_{n}\frac{\left\lbrack {r^{2} + \left( {a_{n}^{\prime}R_{0}} \right)^{2}} \right\rbrack^{1/2} - {a_{n}^{\prime}R_{0}}}{\left\lbrack {r^{2} + \left( {a_{n}^{\prime}R_{0}} \right)^{2}} \right\rbrack^{1/2}}{\overset{\rightarrow}{e}}_{\phi}}}}} & (29)\end{matrix}$

The knowledge of this field distribution allows the calculation of theloss in the conducting sample.

Hyperbolic Tip

For tip-sample separations <R₀, a spherical tip turns out to be anexcellent approximation, but the approximation does not work well fortip-sample separations >R₀. To increase the useful range of thequasistatic model, additional modeling is performed to accurately modelthe electric field over a larger portion of the tip. An exact solutionexists for a given hyperbolic tip at a fixed distance from a conductingplane. The potential is:

$\begin{matrix}{{\phi = {\phi_{0}\frac{\log\left\lbrack \frac{1 + v}{1 - v} \right\rbrack}{\log\left\lbrack \frac{1 + v_{0}}{1 - v_{0}} \right\rbrack}}},} & (30)\end{matrix}$The surface charge/unit area is:

$\quad\begin{matrix}\begin{matrix}{{\sigma = {\frac{1}{4\;\pi}E}}}_{v_{0}} \\{= {{- \frac{1}{4\;\pi}}\frac{2\;\phi_{0}}{a\left( {1 - v_{0}^{2}} \right)}{\frac{1}{\log\left\lbrack \frac{1 + v_{0}}{1 - v_{0}} \right\rbrack}\left\lbrack \frac{1 - v_{0}^{2}}{u^{2} + 1 - v_{0}^{2}} \right\rbrack}^{1/2}}} \\{= {{- \frac{1}{4\;\pi}}\frac{2\;\phi_{0}}{a\left( {1 - v_{0}^{2}} \right)}{\frac{1}{\log\left\lbrack \frac{1 + v_{0}}{1 - v_{0}} \right\rbrack}\left\lbrack \frac{1}{\frac{y^{2}}{{a^{2}\left( {1 - v_{0}^{2}} \right)}^{2}} + 1} \right\rbrack}^{1/2}}}\end{matrix} & (31)\end{matrix}$The area/dy is:

$\begin{matrix}{{y = {{au}\left( {1 - v_{0}^{2}} \right)}^{1/2}}{x = {{av}_{0}\left( {1 + u^{2}} \right)}^{1/2}}{{\frac{\mathbb{d}x}{\mathbb{d}y} = {{av}_{0}{\frac{y}{a^{2}\left( {1 - v_{0}^{2}} \right)}\left\lbrack {1 + \frac{y^{2}}{a^{2}\left( {1 - v_{0}^{2}} \right.}} \right\rbrack}^{{- 1}/2}}},}} & (32) \\{{\frac{\mathbb{d}A}{\mathbb{d}y} = {{2\;\pi\;{y\left( {1 + \left( \frac{\mathbb{d}x}{\mathbb{d}y} \right)^{2}} \right)}^{1/2}} = {2\;\pi\;{y\left\lbrack \frac{1 + \frac{y^{2}}{{a^{2}\left( {1 - v_{0}^{2}} \right)}^{2}}}{1 + \frac{y^{2}}{a^{2}\left( {1 - v_{0}^{2}} \right)}} \right\rbrack}^{1/2}}}},} & (33)\end{matrix}$So, for a given v₀ and a, the charge on a hyperbolic tip between y_(min)and y_(max) is given by:

$\begin{matrix}\begin{matrix}{Q = {\int_{y\; m\; i\; n}^{y\;{ma}\; x}{\sigma\frac{\mathbb{d}A}{\mathbb{d}y}{\mathbb{d}y}}}} \\{= {\frac{\phi_{0}}{a\left( {1 - v_{0}^{2}} \right)}\frac{1}{\log\left\lbrack \frac{1 + v_{0}}{1 - v_{0}} \right\rbrack}{\int_{y\; m\; i\; n}^{y\;{ma}\; x}{\mathbb{d}{{yy}\left\lbrack {1 + \frac{y^{2}}{a^{2}\left( {1 - v_{0}^{2}} \right)}} \right\rbrack}^{{- 1}/2}}}}} \\{= {\phi_{0}a\;{\frac{1}{\log\left\lbrack \frac{1 + v_{0}}{1 - v_{0}} \right\rbrack}\left\lbrack {1 + \frac{y^{2}}{a^{2}\left( {1 - v_{0}^{2}} \right)}} \right\rbrack}^{1/2}\begin{matrix}y_{{ma}\; x} \\y_{m\; i\; n}\end{matrix}}}\end{matrix} & (34)\end{matrix}$Given proper choice of the limits of integration and tip parameters, Eq.34 may be used to more accurately model the tip-sample capacitance asdetailed below.

iii. Large Tip-Sample Separations

The long-range dependence is modeled by calculating C_(tip-sample). Forlarge tip-sample separations, there is no problem. (separationsroughly>tip radius). The capacitance is calculated as the sum of thecontribution from a cone and a spherical tip. For the cone, the chargeonly outside the tip radius is considered. This solution can beapproximately adapted to a variable distance. FIG. 9 describes thetip-sample geometry modeled; where:

Tip radius: R₀

Opening angle: θ

Wire radius: R_(wire)

Tip-sample separation: x₀

To approximate the contribution of the conical portion of the tip to thetip-sample capacitance, the conical portion of the tip is divided into Nseparate portions, each portion n extending from y_(n−1) to y_(n). Foreach portion, the hyperbolic parameters (a, v₀) are found for which thehyperbola intersects and is tangent to the center of the portion.

Given points x, y on a hyperbola, and the slopes, find the hyperbolaintersecting

and tangent to point (x, y).x=av(u ²+1)^(1/2)  (35)y=au(1−v ²)^(1/2)  (36)Eliminate u.

$\begin{matrix}{x = {{av}\left( {1 + \frac{y^{2}}{\left( {1 - v^{2}} \right)a^{2}}} \right)}^{1/2}} & (37) \\{\frac{\mathbb{d}x}{\mathbb{d}y} = {{av}\;\frac{y}{\left( {1 - v^{2}} \right)a^{2}}\left( {1 + \frac{y^{2}}{\left( {1 - v^{2}} \right)a^{2}}} \right)^{{- 1}/2}}} & (38)\end{matrix}$Eliminate a.From Derivative Equation:

$\begin{matrix}{a^{2} = {\frac{y^{2}}{s^{2}}\left( \frac{{v^{2}\left( {1 + s^{2}} \right)} - s^{2}}{\left( {1 - v^{2}} \right)^{2}} \right)}} & (39)\end{matrix}$

Substitute into Equation for Hyperbola:

$\begin{matrix}{{{\frac{x^{2}}{v^{2}} - \frac{y^{2}}{1 - v^{2}}} = {a^{2} = {\frac{y^{2}}{s^{2}}\left( \frac{{v^{2}\left( {1 + s^{2}} \right)} - s^{2}}{\left( {1 - v^{2}} \right)^{2}} \right)}}}{{{x^{2}\left( {1 - v^{2}} \right)}^{2} - {y^{2}{v^{2}\left( {1 - v^{2}} \right)}}} = {{y^{2}\left( {v^{4} - v^{2}} \right)} + \frac{y^{2}v^{4}}{s^{2}}}}} & (40) \\{{v = \frac{xs}{{xs} + y}},} & (41)\end{matrix}$get a from above. Finally, the charges accumulated on each portion ofthe tip are summed and a capacitance is obtained.

The contribution from each conical portion of the tip is givenapproximately by:

$\begin{matrix}{{{C_{n_{{cone} - {sample}}} = {a\;{\frac{1}{\log\left\lbrack \frac{1 + v_{0}}{1 - v_{0}} \right\rbrack}\left\lbrack {1 + \frac{y^{2}}{a^{2}\left( {1 - v_{0}^{2}} \right)}} \right\rbrack}^{1/2}}}}_{y_{n - 1}}^{y_{n}},} & (42)\end{matrix}$where C_(n) _(cone-sample) is the contribution to the tip-samplecapacitance from the nth portion of the cone.

Assuming N equally spaced cone portions, for the sphere+cone tipmodeled,

here:

$y_{n} = {x + R_{0} + {\cot\;{\theta\left( {n\frac{\;{R_{wire} - R_{0}}}{N}} \right)}}}$The total tip-sample capacitance is then given by the sum of theportions of the cone and the spherical portion of the tip,C_(sphere-sample), as given by Eq. 24. (C_(sphere-sample) is substitutedfor C_(tip-sample) to reduce confusion.)

$C_{{tip} - {sample}} = {C_{{sphere} - {sample}} + {\sum\limits_{n = 1}^{N}C_{n_{{cone} - {sample}}}}}$

For small tip-sample separations, this model does not work well. So thecapacitance is calculated using the spherical model (which dominates)and the line tangent to the contribution from the cone is made. FIG. 10shows C_(r) using this approximation.

Embodiment B

The above described models are applied to the regulation of thetip-sample separation for dielectric and conductive materials. Inprinciple, with above models, the relationship between tip sampledistance, electrical impedance and measured signals (f_(r) and Q asfunction of sample difference, bias fields and other variables) is knownprecisely, at least when the tip is very close to the sample. Ifmeasured f_(r) and Q signal points (and their derivatives with respectto electric or magnetic fields, distance and other variables) are morethan unknown parameters, the unknowns can be uniquely solved. If bothtip-sample distance and electrical impedance can be determinedsimultaneously, then the tip-sample distance can be easily controlled,so that the tip is always kept above the sample surface with a desiredgap (from zero to microns). Both topographic and electrical impedanceprofiles can be obtained. The calculation can be easily performed bydigital signal processor or any computer in real time or after the dataacquisition.

Since the physical properties are all calculated from the f_(r) and Qand their derivatives, the temperature stability of the resonator iscrucial to ensure the measurement reproducibility. The sensitivity verymuch depends on the temperature stability of the resonator. Effort todecrease the temperature variation of resonator using lowthermal-coefficient-ceramic materials to construct the resonator shouldbe useful to increase the sensitivity of the instrument.

d) Tip-Sample Distance Control for Dielectric Materials

i.) For Samples of Constant Dielectric Constant, the Tip-SampleSeparation can be Regulated by Measurement of f_(r).

Other physical properties, i.e. nonlinear dielectric or loss tangent canbe measured simultaneously.

FIG. 11 illustrates the operation of the microscope for simultaneousmeasurement of the topography and nonlinear dielectric constant. Fromthe calibration curve of resonant frequency versus tip-sampleseparation, a reference frequency fret is chosen to correspond to sometip-sample separation. To regulate the tip-sample distance, aphase-locked loop (FIG. 11) is used. A microwave signal of frequencyfret is input into the cavity 10, with the cavity output being mixedwith a signal coming from a reference path. The length of the referencepath is adjusted so that the mixed output is zero when the resonancefrequency of the cavity matches f_(ref). The output of the phasedetector 41 is fed to an integrator 44, which regulates the tip-sampledistance by changing the extension of a piezoelectric actuator 50(Burleigh PZS-050) to maintain the integrator output near zero. Forsamples with uniform dielectric constant, this corresponds to a constanttip-sample separation. To measure ∈₃₃₃, an oscillating voltage V_(Ω), offrequency f_(Ω), is applied to the silver backing of the sample and theoutput of the phase detector is monitored with a lock-in amplifier (SR830). This bias voltage will modulate the dielectric constant of anonlinear dielectric material at f_(Ω). Since f_(Ω) exceeds the cut-offfrequency of the feedback loop, the high frequency shift in c from V_(Ω)does not affect the tip-sample separation directly. By measuring theapplied voltage to the piezoelectric actuator and the first harmonicvariation in the phase output simultaneously, sample topography and ∈₃₃₃can be measured simultaneously.

FIG. 12 shows images of topography and for a periodically poledsingle-crystal LiNbO₃ wafer. The crystal is a 1 cm×1 cm single crystalsubstrate, poled by application of a spatially periodic electric field.The poling direction is perpendicular to the plane of the substrate. Thetopographic image is constructed by measuring the voltage applied to thepiezoelectric actuator. It is essentially featureless, with theexception of a constant tilt and small variations in height correlatedwith the alternating domains. The small changes are only observable ifthe constant tilt is subtracted from the figure. Since ∈_(ijk) is athird-rank tensor, it reverses sign when the polarization switches,providing an image of the domain structure. The nonlinear image isconstructed by measuring the first harmonic of the variation in outputof the phase detector using a lock-in amplifier. Since ∈_(ijk) reverseswhen the polarization switches, the output of the lock-in switches signwhen the domain direction switches. The value (−2.4×10⁻¹⁹ F/N) is within20% of bulk measurements. The nonlinear image clearly shows thealternating domains.

Ferroelectric thin films, with their switchable nonvolatilepolarization, are also of great interest for the next generation ofdynamic random access memories. One potential application of thisimaging method would be in a ferroelectric storage media. A number ofinstruments based on the atomic force microscope have been developed toimage ferroelectric domains either by detection of surface charge or bymeasurement of the piezoelectric effect. The piezoelectric effect, whichis dependent on polarization direction, can be measured by applicationof an alternating voltage and subsequent measurement of the periodicvariation in sample topography. These instruments are restricted totip-sample separations less than 10 nanometers because they rely oninteratomic forces for distance regulation, reducing the possible datarate. Since the inventive microscope measures variations in thedistribution of an electric field, the tip-sample separation can beregulated over a wide range (from nanometers to microns).

ii.) For Samples with Varying Dielectric Constant, f_(r) Changes with ∈.

To extract the dielectric constant and topography simultaneously, anadditional independent signal is required. This can be accomplished inseveral ways:

1. Measuring more than one set of data for f_(r) and Q at differenttip-sample distances. This method is especially effective when thetip-sample distance is very small. The models described in Embodiment Acan then be used to determine the tip-sample distance and electricalimpedance through DSP or computer calculation. In this approach, as thetip is approaching the sample surface the DSP will fit the tip-sampledistance, dielectric constant and loss tangent simultaneously. Thesevalues should converge as the tip-sample distance decreases. Therefore,this general approach will provide a true non-contact measurement mode,as the tip can kept at any distance away from the sample surface as longas the sensitivity (increase as tip-sample distance decreases) is enoughfor the measurement requirement. This mode is referred to as non-contacttapping mode. During the scanning, at each pixel the tip is first pullback to avoid crash before lateral movement. Then the tip will approachthe surface as DSP calculate the dielectric properties and tip-sampledistance. As the measurement value converge to have less error thanspecified or calculated tip-sample distance is less than a specifiedvalue, the tip stop approaching and DSP record the final values for thatpixel. In this approach, a consistent tip-sample moving element iscritical, i.e. the element should have a reproducible distance vs, e.g.control voltage. Otherwise, it increases the fitting difficulty andmeasurement uncertainty. This requirement to the z-axis moving elementmay be hard to satisfy. An alternative method is to independently encodethe z-axis displacement of the element. Capacitance sensor and opticalinterferometer sensor or any other distance sensor can be implemented toachieve this goal.

2. In particular, when the tip is in soft contact (only elasticdeformation is involved) to the surface of the sample, there will be asharp change in the derivatives of signals (fr and Q) as function ofapproaching distance. This method is so sensitive that it can be used todetermine the absolute zero of tip-sample distances without damaging thetip. Knowing the absolute zero is very useful and convenient for furtherfitting of the curve to determine the tip-sample distance and electricalimpedance. A soft contact “tapping mode” (as described in above) can beimplemented to perform the scan or single point measurements. Theapproaching of tip in here can be controlled at any rate by computer orDSP. It can be controlled interactively, i.e. changing rate according tothe last measurement point and calculation.

3. The method described in 1) can be alternatively achieved by a fixedfrequency modulation in tip-sample distance and detected by a lock-inamplifier to reduce the noise. The lock-in detected signal will beproportional to the derivative of fr and Q as function of tip-sampledistance. A sharp decrease in this signal can be used as a determinationof absolute zero (tip in soft contact with sample surface withoutdamaging the tip). Using relationships described in Embodiment A, anydistance of tip-sample can be maintained within a range that theserelationship is accurate enough.

Details

At a single tip-sample separation, the microwave signal is determined bythe dielectric constant of the sample. However, the microwave signal isa fraction of both the tip-sample separation and the dielectricconstant. Thus, the dielectric constant and tip-sample separation can bedetermined simultaneously by the measurement of multiple tip-sampleseparations over a single point. Several methods can be employed toachieve simultaneous measurement of tip-sample separation and dielectricconstant. First, the derivative of the tip-sample separation can bemeasured by varying the tip-sample separation. Given a model of thetip-sample capacitance, (Eqn. 3), the tip-sample separation anddielectric constant can then be extracted. The dependence on tip-sampleseparation and dielectric constant can be modeled using a modified fermifunction.

$\begin{matrix}{C_{r} = {4\pi\; ɛ_{0}R\;\frac{{\ln\;{\left( {1 - b} \right)/b}} + 1}{{\exp\left\{ {{G(ɛ)}\left\lbrack {{\ln\; a^{\prime}} - {x_{0}(ɛ)}} \right\rbrack} \right\}} + 1}}} & (43)\end{matrix}$where b=(∈−∈₀)/(∈+∈₀). Furthermore, G(∈) and x₀(∈) can be fitted wellwith rational functions as:

$\begin{matrix}\left\{ \begin{matrix}{{G(ɛ)} = \frac{{9.57 \times 10^{- 1}} + {2.84 \times 10^{- 2}ɛ} + {3.85 \times 10^{- 5}ɛ^{2}}}{1 + {4.99 \times 10^{- 2}ɛ} + {1.09 \times 10^{- 4}ɛ^{2}}}} \\{{x_{0}(ɛ)} = \frac{{5.77 \times 10^{- 1}} + {1.31 \times 10^{- 1}ɛ} + {3.55 \times 10^{- 4}ɛ^{2}}}{1 + {3.68 \times 10^{- 2}ɛ} + {5.16 \times 10^{- 5}ɛ^{2}}}}\end{matrix} \right. & (44)\end{matrix}$FIG. 13 illustrates the agreement between Eqn. 3 and Eqn. 43.

Equation 43 is suitable for rapid calculation of the tip-sampleseparation and dielectric constant. The tip-sample separation anddielectric constant can also be extracted by construction of a look-uptable. The architecture described in f) below is then used to regulatethe tip-sample separation. FIG. 14 a and FIG. 14 b show the variation ofthe derivative signal versus tip-sample variation. FIG. 14 a is modeledassuming a 10 μm spherical tip. FIG. 14 b is a measured curve.Maintaining the tip-sample separation at the maximum point of thederivative signal can also regulate the tip-sample separation. This hasbeen demonstrated by scanning the tip-sample separation and selectingthe zero slope of the derivative signal. It can also be accomplished byselecting the maximum of the second harmonic of the microwave signal.

e) Tip-Sample Distance Control for Conductive Materials

For conductive materials, the tip-sample separation and microwaveresistivity can be measured simultaneously in a similar fashion. SinceEqn. 24 is independent of conductivity for good metals, C_(tip-sample)can be used as a distance measure and control. This solution should begenerally applicable to a wide class of scanned probe microscopes thatinclude a local electric field between a tip and a conducting sample. Itshould prove widely applicable for calibration and control ofmicroscopes such as scanning electrostatic force and capacitancemicroscopes.

From the calibration curves, a frequency f_(ref) is chosen to correspondto some tip sample separation (FIG. 15). The tip-sample separation isthen regulated to maintain the cavity resonance frequency at f_(ref).This can be accomplished digitally through the use of the digital signalprocessor described in f) below. An analog mechanism can and has alsobeen used. A phase-locked loop described in FIG. 11 has also been usedto regulate the tip-sample separation. FIG. 16( a)(c) illustrate themeasurement of topography with constant microwave conductivity. FIG. 16(b)(d) illustrate the measurement of conductivity variations.

This method allows submicron imaging of the conductivity over largelength scales. This method has the advantage of allowing distanceregulation over a wide length scale (ranging from microns to nanometers)giving rise to a capability analogous to the optical microscope'sability to vary magnification over a large scale.

Force Sensor Distance Feedback Control

In many cases, an absolute and independent determination ofzero-distance is desirable as other methods can rely on modelcalculations and require initial calibration and fitting. A method thatrelies on measuring the vibration resonant frequency of the tip as aforce sensor has been implemented. When the tip approaches the samplesurface, the mechanical resonant frequency of the tip changes. Thischange can be used to control the tip-sample distance. This approach hasbeen shown to be feasible and that this effect does exist. This effecthas been found to exist over a very long range (˜1 micron). Thelong-range effect is believed to come from the electrostatic force andthe short-range effect is from a shear force or an atomic force. A lowfrequency DDS-based digital frequency feedback control electronicssystem similar to the microwave one discussed above is implemented totrack the resonant frequency and Q of this mechanical resonator. Themeasure signals will then be used to control the distance. This issimilar to shear force measurement in near field scanning microscope(NSOM). The signal is derived from the microwave signal, not fromseparate optical or tuning fork measurements as in other methods. Thisfeature is important since no extra microscope components are required.The electronics needed is similar to the high frequency case. Thisfeatures is critical for high-resolution imaging and accuratecalibration of the other two methods.

Integration of AFM

It is very useful to integrate an atomic force microscope (AFM) tip withSEMM in some applications where a high-resolution topography image isdesirable. The tip resonant frequency (10 kHz) and quality factor limitthe bandwidth of shear force feedback in closed loop applications.Previously, in a scanning capacitance microscope (SCM), an AFM tip hasbeen connected to a microwave resonator sensor to detect the change incapacitance between the tip and the bottom electrode. In principle, SCMis similar to our SEMM. However, insufficient coupling between the tipand resonator, very large parasitic capacitance in the connection, andlack of shielding for radiation prevented the SCM from having anysensitivity in direct (dc mode) measurements of the tip-samplecapacitance without a bottom electrode and ac modulation. (SCM only canmeasure dC/dV).

A new inventive design will allow easy integration of an AFM tip to theSEMM without losing the high sensitivity experienced with the SEMM. Thisnew design is based on a modification stripline resonator with theinventors' proprietary tip-shielding structure (see FIG. 21). Since theAFM tip is connected within less than 1 mm of the central strip line,the sensitivity will not be reduced seriously and parasitic capacitancewith be very small (shielding also reduces the parasitic capacitance).The AFM tip is custom designed to optimize the performance. With thisunique inventive design, nanometer resolution can be achieved in bothtopography and electrical impedance imaging, which is critical in gateoxide doping profiling and many other applications.

Embodiment C

By contrast to most types of microscope, SEMM measures a complexquantity, i.e., the real and imaginary parts of the electricalimpedance. This is realized by measuring the changes in the resonantfrequency (f_(r)) and quality factor (Q) of the resonatorsimultaneously. A conventional method of measuring these two quantitiesis to sweep the frequency of the microwave generator and measure theentire resonant curve. For each measurement, this can take seconds tominutes depending on the capabilities of the microwave generator. Thesemeasurements are limited by the switching speed of a typical microwavegenerator to roughly 20 Hz. With the use of a fast direct digitalsynthesizer based microwave source, the throughput can be improved toroughly 10 kHz, but is still limited by the need to switch over a rangeof frequencies. Another method is to implement an analog phase-lockedloop for frequency feedback control. This method can track the changingresonant frequency in real time and measure f_(r) and Q quickly.However, one has to use a voltage-controlled-oscillator (VCO) as amicrowave generator which usually only has a frequency stability of10⁻⁴. This low frequency stability will seriously degrade thesensitivity of the instrument. Since Δ∈/∈˜500Δf_(r)/f_(r), frequencyinstability in the VCO will limit measurement accuracy. Another problemis that interaction between this frequency feedback loop and thetip-sample distance feedback loop can cause instability and oscillation,which will seriously limit the data rate.

A direct digital synthesizer (DDS) based microwave generator is used toimplement the method according to the present invention. In a preferredembodiment, the DSS has a frequency stability of better than 10⁻⁹. Theinventive method fixes the frequency of the microwave signal at theprevious resonant frequency and measures I/Q signals simultaneously.Since the microwave frequency is fixed, the DDS switching speed does notlimit the data rate. By measuring the in-phase and quadrature microwavesignals, the inventors can derive f_(r) and Q. Near resonance, thein-phase and quadrature signals are given by:i=A sin θ

Q=A cos θ, where A is the amplitude of the microwave signal on resonanceand θ is the phase shift of the transmitted wave. Given i, q, thecurrent input microwave frequency, and the input coupling constants, thecurrent f_(r) and Q can be calculated. For a resonator with initialquality factor Q₀, transmitted power A₀, and resonant frequency f₀,driven at frequency f,

${{\Delta\; f} = {{\frac{1}{2}\tan\;\theta\;\frac{fr}{Q}}\mspace{34mu} = {\frac{1}{2}c\mspace{11mu}\frac{\sin\;\theta}{A}}}},{{fr} = {f - {\Delta\; f}}}$${Q = {\frac{1}{c}{fr}\;\frac{A}{\cos\;\theta}}},$where

$c = \frac{A \circ f \circ}{Q \circ}$Then, f_(r) and Q can be obtained by

${fr} = {f - {\frac{1}{2}c\;\frac{i}{i^{2} + q^{2}}}}$$Q = {\frac{1}{c}f_{r}\frac{i^{2} + q^{2}}{q}}$

Initially, since the I/Q mixer does not maintain perfect phase oramplitude balance, these quantities are calibrated. To calibrate therelative amplitudes at a given frequency of the i and q outputs of themixer, the relative values of the outputs are measured when thereference signal is shifted by 90 degrees. This can easily be extendedby means of a calibration table.

To calibrate the relative phases of the i and q outputs at a givenfrequency, the i output of the mixer is measured on resonance. Atresonance,

i/q=δ, where δ is the phase error of the I/Q mixer.

Near Resonance,

$\begin{matrix}{i = {A\;\sin\;\left( {\theta + \delta} \right)}} \\{= {{A\;\sin\;\theta\;\cos\;\delta}\; + {A\;\sin\;\delta\;\cos\;\theta}}} \\{= {{A\;\sin\;\theta} + {A\;\delta\;\cos\;\theta}}} \\{= {{A\;\sin\;\theta}\; + {q\;\delta}}}\end{matrix}$This allows the correction of the phase error of the mixer.

This method of measurement only requires one measurement cycle.Therefore, it is very fast and limited only by the DSP calculationspeed. To increase the working frequency range, the DSP is used tocontrol the DDS frequency to shift when the resonant frequency change isbeyond the linear range. This method allows data rates around 100 kHz-1MHz (limited by the bandwidth of the resonator) and frequencysensitivity below 1 kHz

$\left( {\frac{\Delta\; f_{r}}{f_{r}} \approx {10^{- 6}\mspace{20mu}{to}\mspace{14mu} 10^{- 7}}} \right).$

f) Data Acquisition and Control Electronics

FIGS. 17 and 18 illustrate the architecture of the inventive dataacquisition and control electronics. FIG. 17 contains a schematic forthe EMP. FIG. 18 is a flow chart describing the operation of the SEMP.

To eliminate the communication bottleneck between data acquisition,control electronics and the computer, the high performance PCI bus isadopted for every electronic board. A main board with four high-speeddigital signal processors (DSPs) is used to handle data acquisition,feedback control loops and other control functions separately. Fourinput data signals (A/Ds) and six control signals (D/As) wereimplemented.

The input signals include:

1) in-phase (I=A sin θ, where A is the amplitude and θ the phase)signal,

2) quadrature (q=A cos θ) signal,

3) in-phase signal of tip vibration resonant frequency or tapping modesignal electric/magnetic/optical field modulations,

4) quadrature signal of tip vibration resonance,

The output signals include:

1-3) fine x-y-z piezo-tube control signals,

4) z-axis coarse piezo-step-motor signal,

5-6) coarse x-y stage signals.

The DSPs are dedicated to specific functions as follows:

DSP1: microwave frequency f_(r) and Q data acquisition,

DSP2: tip mechanical resonant frequency f_(m) and Q_(m) or microwavederivative signal data acquisition,

DSP3: tip-sample distance feedback control,

DSP4: x-y fine/coarse scan and image data acquisition,

The division of labor between a number of fast processors simplifiesdesign and allows rapid processing of multiple tasks. A fast system busis necessary to allow rapid transfer of data to the display and betweenneighboring boards.

Stepping Motor—Coarse Positioning

Embodiment D

For all scanned probe microscopes, increasing resolution decreases themeasurable scan range. Microscopes must be able to alter their scanposition by millimeters to centimeters while scanning with highresolution over hundreds of microns. Given samples of macroscopic scale,means must exit to adjust the tip-sample separation over macroscopic(mm) distances with high stability over microscopic distances (nm).Conventional piezoelectric positioners are capable only of movements inthe range of hundreds of microns and are sensitive to electronic noiseeven when stationary. In addition, they are subject to large percentagedrifts. (>1% of scan range) As such, separate means of coarse and fineadjustment of position are necessary. A coarse approach inventivemechanism by means of a novel piezoelectric stepper motor has beendesigned to accomplish this.

FIG. 19 illustrates the design and operation of the stepper motor. Thecross-sectional view illustrates that the motor consists of a sapphireprism in the form of an equilateral triangle clamped into an outercasing. There are 3 Piezoelectric stacks topped by a thin sapphire platecontact each side of the prism. Each piezoelectric stack consists of alower expansion plate, which is used to grip and release the prism, anupper shear plate, which is used to move the prism, and a thin sapphireplate, which is used to provide a uniform surface. FIG. 20 shows thesequence of motion. At step (a), the motor is stationary. At step (b), asmooth rising voltage is applied to each shear place and the centerprism moves. At step (c), the voltage to the expansion plates labeled(2) is reduced. This reduces the pressure applied by those plates andthus the frictional force. The center points the motor fixed. At step(d), the voltage to the shear plates labeled (1) is reduced. Since theplates have been retracted, the friction between these plates and theprism is reduced, letting the center plates hold the prism. At step (e),the voltage to the expansion plates labeled (2) is increased, increasingthe pressure applies by the other plates. At step (F), the voltage tothe expansion plates labeled (4) is reduced. The outer points hold themotor fixed. At step (g), the voltage to the shear plates labeled (3) isreduced. At step (h), the voltage to the expansion plates labeled (4) isincreased, restoring the pressure applied by the center points to itsoriginal value. This motor has a number of advantages by comparison toearlier designs. Prior designs had only 2 piezoelectric elements perside and relied on a stick-slip motion, similar to pulling a tableclothfrom under a wineglass. The use of a third piezoelectric element on eachside and the addition of an expansion piezoelectric have severalbenefits. First, since the method of motion does not involve slip-stick,it is less prone to vibration induced by the necessary sharp motions.Second, the requirements for extreme cross-sectional uniformity of thecentral element are reduced by the use of the expansion piezoelectricplates.

While the present invention has been particularly described with respectto the illustrated embodiment, it will be appreciated that variousalterations, modifications and adaptations may be made based on thepresent disclosure, and are intended to be within the scope of thepresent invention. While the invention has been described in connectionwith what is presently considered to be the most practical and preferredembodiments, it is to be understood that the present invention is notlimited to the disclosed embodiments but, on the contrary, is intendedto cover various modifications and equivalent arrangements includedwithin the scope of the appended claims.

TABLE 1 Single Crystal Measurement Measured Reported Measured ReportedMaterial ε_(r) ε_(r) tanδ tanδ YSZ 30.0 29 1.7 × 10⁻³ 1.75 × 10⁻³ LaGaO₃ 23.2 25 1.5 × 10⁻³ 1.80 × 10⁻³  CaNdAlO₄ 18.2 ~19.5 1.5 × 10⁻³0.4-2.5 × 10⁻³    TiO₂ 86.8 85 3.9 × 10⁻³  4 × 10⁻³ BaTiO₃ 295 300 0.470.47 YAlO₃ 16.8 16 — 8.2 × 10⁻⁵ SrLaAlO₄ 18.9 20 — — LaALO₃ 25.7 24 —2.1 × 10⁻⁵ MgO 9.5 9.8 — 1.6 × 10⁻⁵ LiNbO₃ 32.0 30 — — (X-cut)

TABLE 2 @Interdigital SEMM (1 GH_(z)) Electrodes (1 GH_(z)) Films ε_(r)tanδ ε_(r) tanδ Ba_(0.7)Sr_(0.3)TiO₃ 707 0.14 750 0.07Ba_(0.5)Sr_(0.5)TiO₃ 888 0.19 868 0.10 SrTiO₃ 292 0.02 297 0.015Ba_(0.24)Sr_(0.35)Ca_(0.41)TiO₃ 150 0.05 *Ba_(0.25)Sr_(0.35)Ca_(0.4)TiO₃240 0.05 @Measurement by S. Kirchoefer and J. Pond, NRL also consistentwith results by NIST group *Film made by H. Jiang and V. Fuflyigin, NZApplied Technology Loss values are higher since we are more sensitive tosurface.

1. A method for using a scanning evanescent microwave probe to determineelectrical properties of a sample, said probe having a tip extendingfrom a coaxial or transmission line resonator, comprising: measuringvariation in resonant frequency and quality factor of said resonatorresulting from interaction of said tip and said sample; wherein saidtip-sample interaction appears as equivalent complex tip-samplecapacitance; and wherein said effective complex tip-sample capacitance,C_(tip-sample) is determined according to C_(tip-sample) =C_(r)+C_(i),wherein C_(r) and C_(i) are the real and imaginary components of thetip-sample capacitance, respectively,${\frac{\Delta\; f}{f_{0}} = {- \frac{Cr}{2\; C}}},{{\Delta\;\left( \frac{1}{Q} \right)} = {{- \left( {\frac{1}{Q} + \frac{2C_{i}}{C_{r}}} \right)}\frac{\Delta\; f}{f_{0}}}},{{\Delta\; f} = {f_{r} - f_{0}}},{{\Delta\;\left( \frac{1}{Q} \right)} = {\frac{1}{Q} - \frac{1}{Q_{0}}}},$and f₀ and Q₀ are the unloaded resonant frequency and quality factor,respectively.
 2. A method as recited in claim 1, wherein said measuringof said variation in resonant frequency and quality factor of saidresonator comprises: obtaining signal from an I/Q mixer; and determiningresonant frequency and quality factor as a function of said signals fromsaid I/Q mixer.
 3. A method as recited in claim 1, further comprising:measuring probe parameters selected from the group consisting ofresonant frequency shift and quality factor shift, wherein the resonantfrequency shift and the quality factor shift results from an interactionbetween the sample and an evanescent electromagnetic field emitted fromsaid probe.
 4. The method for measuring an electromagnetic propertyaccording to claim 3, wherein the measurement is made using quasistaticapproximation modeling.
 5. A method for using a scanning evanescentmicrowave probe to determine electrical properties of a sample, saidprobe having a tip extending from a coaxial or transmission lineresonator, comprising: measuring variation in resonant frequency andquality factor of said resonator resulting from interaction of said tipand said sample; said probe having said tip extending from a microwavecavity; positioning said sample outside said microwave cavity andadjacent said tip; causing said tip to emit an evanescentelectromagnetic field; scanning a surface of said sample with said tipto measure resonant frequency shift of said probe, wherein said resonantfrequency shift results from interaction between said sample and saidevanescent electromagnetic field; and determining said electricalproperties and topography of said sample using the measured resonantfrequency shift.
 6. A method for using a scanning evanescent microwaveprobe to determine electrical properties of a sample, said probe havinga tip extending from a coaxial or transmission line resonator,comprising: measuring variation in resonant frequency and quality factorof said resonator resulting from interaction of said tip and saidsample; said probe having said tip extending from a microwave cavity;positioning said sample outside said microwave cavity and adjacent saidtip; causing said tip to emit an evanescent electromagnetic field;measuring a quality factor shift of said probe, wherein said qualityfactor shift results from interaction between said sample and saidevanescent electromagnetic field; and determining said electricalimpedance and the distance between said tip and said sample using themeasured quality factor shift.
 7. A method as recited in claim 6,wherein said probe comprises a scanning evanescent microwave probehaving said tip extending from a coaxial or transmission line resonator.8. A method as recited in claim 6, wherein said measurements ofelectrical impedance are selected from the group consisting essentiallyof quantitative and qualitative measurements.
 9. A method as recited inclaim 6, wherein said electrical impedance comprises complex dielectricconstant and conductivity of said sample.
 10. A method as recited inclaim 6, wherein said sample comprises a material selected from thegroup consisting essentially of dielectric insulators, semiconductors,metallic conductors and superconductors.
 11. A method as recited inclaim 6, wherein said sample comprises a multi-layered material.
 12. Amethod as recited in claim 11, wherein said sample comprises a materialselected from the group consisting essentially of dielectric insulators,semiconductors, metallic conductors and superconductors.
 13. A method asrecited in claim 6, wherein said tip-sample interaction is measured witha modulated external field applied to a backing of said sample.
 14. Amethod as recited in claim 13, further comprising detecting thederivatives of the resonant frequency or phase, quality factor oramplitude of said probe with respect to modulation of said externalfield using a lock-in amplifier having an operating frequency coherentwith the frequency of the modulation.
 15. A method as recited in claim13, wherein said external field comprises a bias electric field.
 16. Themethod as recited in claim 6, wherein the measurement is made underquasistatic approximation conditions.
 17. A method for using a scanningevanescent microwave probe to determine electrical properties of asample, said probe having a tip extending from a coaxial or transmissionline resonator, comprising: measuring variation in resonant frequencyand quality factor of said resonator resulting from interaction of saidtip and said sample; said probe having said tip extending from amicrowave cavity; positioning said sample outside said microwave cavityand adjacent but not in contact with said tip; causing said tip to emitan evanescent electromagnetic field; scanning a surface of said samplewith said tip to measure quality factor shift of said probe, whereinsaid quality factor shift results from interaction between said sampleand said evanescent electromagnetic field; and determining electricalproperties and topography of said sample using the measured qualityfactor shift.